2: Lead — Wannier-interpolated Fermi surface¶

Outline: Obtain MLWFs for the four lowest states in lead. Use Wannier interpolation to plot the Fermi surface.
Generation Details: From pwscf, using norm-conserving pseudopotentials and a
4\(\times\)4\(\times\)4 k-point grid. Starting guess: atom-centred sp\(^3\) hybrid orbitals
Directory: tutorials/tutorial02/ Files can be downloaded from here
Input Files
- lead.win The master input file
- lead.mmn The overlap matrices \(\mathbf{M}^{(\mathbf{k},\mathbf{b})}\)
- lead.amn Projection \(\mathbf{A}^{(\mathbf{k})}\) of the Bloch states onto a set of trial localised orbitals
- lead.eig The Bloch eigenvalues at each k-point. For interpolation only

The four lowest valence bands in lead are separated in energy from the higher conduction states (see bandstructure plot). The MLWFs of these states have partial occupancy. MLWFs describing only the occupied states would be poorly localised.

Run wannier90 to minimise the MLWFs spread
Terminal
```
wannier90.x lead
```
Inspect the output file lead.wout.
Use Wannier interpolation to generate the Fermi surface of lead. Rather than re-running the whole calculation we can use the unitary transformations obtained in the first calculation and restart from the plotting routine. Add the following keywords to the lead.win file:
Input file
```
restart = plot
fermi_energy = 5.2676
fermi_surface_plot = true
```
and re-run wannier90. The value of the Fermi energy (5.2676 eV) was obtained from the initial first principles calculation. wannier90 calculates the band energies, through

interpolation, on a dense mesh of k-points in the Brillouin zone. The density of this grid is controlled by the keyword fermi_surface_num_points. The default value is 50 (i.e., 50\(^3\) points). The Fermi surface file lead.bxsf can be viewed using XCrySDen, e.g.,
Terminal
```
xcrysden --bxsf lead.bxsf
```

Image title — Bandstructure of lead showing the position of the Fermi level. Only the lowest four bands are included in the calculation.